5.50. change_continuity

DESCRIPTIONLINKSGRAPHAUTOMATON
Origin

N.Β Beldiceanu

Constraint

πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’π™½π™±_π™Ώπ™΄πšπ™Έπ™Ύπ™³_𝙲𝙷𝙰𝙽𝙢𝙴,𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ,𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴,π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴,𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ,π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ,𝙽𝙱_𝙲𝙷𝙰𝙽𝙢𝙴,𝙽𝙱_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,π™²πšƒπš

Arguments
𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™·π™°π™½π™Άπ™΄πšπšŸπšŠπš›
𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™ΈπšƒπšˆπšπšŸπšŠπš›
𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™·π™°π™½π™Άπ™΄πšπšŸπšŠπš›
π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_π™²π™·π™°π™½π™Άπ™΄πšπšŸπšŠπš›
𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™ΈπšƒπšˆπšπšŸπšŠπš›
π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™ΈπšƒπšˆπšπšŸπšŠπš›
𝙽𝙱_π™²π™·π™°π™½π™Άπ™΄πšπšŸπšŠπš›
𝙽𝙱_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™ΈπšƒπšˆπšπšŸπšŠπš›
πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›-πšπšŸπšŠπš›)
π™²πšƒπšπšŠπšπš˜πš–
Restrictions
𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_𝙲𝙷𝙰𝙽𝙢𝙴β‰₯0
𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆβ‰₯0
𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴β‰₯0
π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴β‰₯𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴
𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆβ‰₯0
π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆβ‰₯𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ
𝙽𝙱_𝙲𝙷𝙰𝙽𝙢𝙴β‰₯0
𝙽𝙱_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆβ‰₯0
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πšŸπšŠπš›)
π™²πšƒπšβˆˆ[=,β‰ ,<,β‰₯,>,≀]
Purpose

On the one hand a change is defined by the fact that constraint πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš› π™²πšƒπš πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i+1].πšŸπšŠπš› holds.

On the other hand a continuity is defined by the fact that constraint πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš› π™²πšƒπš πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i+1].πšŸπšŠπš› does not hold.

A period of change on variables

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš›,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i+1].πšŸπšŠπš›,...,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[j].πšŸπšŠπš› (i<j)

is defined by the fact that all constraints πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[k].πšŸπšŠπš› π™²πšƒπš πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[k+1].πšŸπšŠπš› hold for k∈[i,j-1].

A period of continuity on variables

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš›,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i+1].πšŸπšŠπš›,...,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[j].πšŸπšŠπš› (i<j)

is defined by the fact that all constraints πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[k].πšŸπšŠπš› π™²πšƒπš πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[k+1].πšŸπšŠπš› do not hold for k∈[i,j-1].

The constraint πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ holds if and only if:

  • 𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_𝙲𝙷𝙰𝙽𝙢𝙴 is equal to the number of periods of change,

  • 𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ is equal to the number of periods of continuity,

  • 𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴 is equal to the number of variables of the smallest period of change,

  • π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴 is equal to the number of variables of the largest period of change,

  • 𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ is equal to the number of variables of the smallest period of continuity,

  • π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ is equal to the number of variables of the largest period of continuity,

  • 𝙽𝙱_𝙲𝙷𝙰𝙽𝙢𝙴 is equal to the total number of changes,

  • 𝙽𝙱_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ is equal to the total number of continuities.

Example
3,2,2,4,2,4,6,4,πšŸπšŠπš›-1,πšŸπšŠπš›-3,πšŸπšŠπš›-1,πšŸπšŠπš›-8,πšŸπšŠπš›-8,πšŸπšŠπš›-4,πšŸπšŠπš›-7,πšŸπšŠπš›-7,πšŸπšŠπš›-7,πšŸπšŠπš›-7,πšŸπšŠπš›-2,β‰ 

FigureΒ 5.50.1 makes clear the different parameters that are associated with the given example for the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚=〈1,3,1,8,8,4,7,7,7,7,2βŒͺ. We place character | for representing a change and a blank for a continuity. On top of the solution we represent the different periods of change, while below we show the different periods of continuity. The πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint holds since:

  • Its number of periods of change 𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_𝙲𝙷𝙰𝙽𝙢𝙴 is equal to 3 (i.e.,Β the 3 periods depicted on top of FigureΒ 5.50.1),

  • Its number of periods of continuity 𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ is equal to 2 (i.e.,Β the 2 periods depicted below FigureΒ 5.50.1),

  • The number of variables of its smallest period of change 𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴 is equal to 2 (i.e.,Β the number of variables involved in the third period of change 7 2 depicted on top of FigureΒ 5.50.1),

  • The number of variables of the largest period of change π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴 is equal to 4 (i.e.,Β the number of variables involved in the first period of change 1 3 1 8 depicted on top of FigureΒ 5.50.1),

  • The number of variables of the smallest period of continuity 𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ is equal to 2 (i.e.,Β the number of variables involved in the first period 8 8 depicted below FigureΒ 5.50.1),

  • The number of variables of the largest period of continuity π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ is equal to 4 (i.e.,Β the number of variables involved in the second period 7 7 7 7 depicted below FigureΒ 5.50.1),

  • The total number of changes 𝙽𝙱_𝙲𝙷𝙰𝙽𝙢𝙴 is equal to 6 (i.e.,Β the number of occurrences of character | in FigureΒ 5.50.1),

  • The total number of continuities 𝙽𝙱_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ is equal to 4.

Figure 5.50.1. Periods of changes and periods of continuities
ctrs/change_continuity1
Typical
𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_𝙲𝙷𝙰𝙽𝙢𝙴>0
𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ>0
𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴>0
𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ>0
𝙽𝙱_𝙲𝙷𝙰𝙽𝙢𝙴>0
𝙽𝙱_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ>0
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>1
πš›πšŠπš—πšπšŽ(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš›)>1
Symmetry

One and the same constant can be added to the πšŸπšŠπš› attribute of all items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.

Remark

If the variables of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ have to take distinct values between 1 and the total number of variables, we have what is called a permutation. In this case, if we choose the binary constraint <, then π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴 gives the size of the longest run of the permutation; A run is a maximal increasing contiguous subsequence in a permutation.

See also

common keyword: πšπš›πš˜πšžπš™, πšπš›πš˜πšžπš™_πšœπš”πš’πš™_πš’πšœπš˜πš•πšŠπšπšŽπš_πš’πšπšŽπš–, πšœπšπš›πšŽπšπšŒπš‘_πš™πšŠπšπš‘Β (timetabling constraint).

Keywords

characteristic of a constraint: automaton, automaton with counters.

combinatorial object: sequence, run of a permutation, permutation.

constraint network structure: sliding cyclic(1) constraint network(2), sliding cyclic(1) constraint network(3).

constraint type: timetabling constraint.

final graph structure: connected component, apartition, acyclic, bipartite, no loop.

miscellaneous: obscure.

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚

Arc generator
π‘ƒπ΄π‘‡π»β†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2)

Arc arity
Arc constraint(s)
πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1.πšŸπšŠπš› π™²πšƒπš πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš›
Graph property(ies)
β€’ 𝐍𝐂𝐂=𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_𝙲𝙷𝙰𝙽𝙢𝙴
β€’ 𝐌𝐈𝐍_𝐍𝐂𝐂=𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴
β€’ πŒπ€π—_𝐍𝐂𝐂=π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴
β€’ 𝐍𝐀𝐑𝐂=𝙽𝙱_𝙲𝙷𝙰𝙽𝙢𝙴

Graph class
β€’ π™°π™²πšˆπ™²π™»π™Έπ™²
β€’ π™±π™Έπ™Ώπ™°πšπšƒπ™Έπšƒπ™΄
β€’ 𝙽𝙾_𝙻𝙾𝙾𝙿

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚

Arc generator
π‘ƒπ΄π‘‡π»β†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2)

Arc arity
Arc constraint(s)
πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1.πšŸπšŠπš›Β¬ π™²πšƒπš πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš›
Graph property(ies)
β€’ 𝐍𝐂𝐂=𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ
β€’ 𝐌𝐈𝐍_𝐍𝐂𝐂=𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ
β€’ πŒπ€π—_𝐍𝐂𝐂=π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ
β€’ 𝐍𝐀𝐑𝐂=𝙽𝙱_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ

Graph class
β€’ π™°π™²πšˆπ™²π™»π™Έπ™²
β€’ π™±π™Έπ™Ώπ™°πšπšƒπ™Έπšƒπ™΄
β€’ 𝙽𝙾_𝙻𝙾𝙾𝙿

Graph model

We use two graph constraints to respectively catch the constraints on the period of changes and of the period of continuities. In both case each period corresponds to a connected component of the final graph.

PartsΒ (A) andΒ (B) of FigureΒ 5.50.2 respectively show the initial and final graph associated with the first graph constraint of the Example slot.

Figure 5.50.2. Initial and final graph of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuityActrs/change_continuityB
(a) (b)
Automaton

FiguresΒ 5.50.3 ,Β 5.50.4 ,Β 5.50.7 ,Β 5.50.8 ,Β 5.50.11 ,Β 5.50.12 andΒ 5.50.15 depict the automata associated with the different graph parameters of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint. For the automata that respectively compute 𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_𝙲𝙷𝙰𝙽𝙢𝙴, 𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ 𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴, 𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴, π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ 𝙽𝙱_𝙲𝙷𝙰𝙽𝙢𝙴 and 𝙽𝙱_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ we have a 0-1 signature variable πš‚ i for each pair of consecutive variables (πš…π™°πš i ,πš…π™°πš i+1 ) of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚. The following signature constraint links πš…π™°πš i , πš…π™°πš i+1 and πš‚ i : πš…π™°πš i π™²πšƒπš πš…π™°πš i+1 β‡”πš‚ i .

Figure 5.50.3. Automaton for the 𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_𝙲𝙷𝙰𝙽𝙢𝙴 parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
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Figure 5.50.4. Automaton for the 𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
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Figure 5.50.5. Hypergraph of the reformulation corresponding to the automaton of the 𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_𝙲𝙷𝙰𝙽𝙢𝙴 parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
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Figure 5.50.6. Hypergraph of the reformulation corresponding to the automaton of the 𝙽𝙱_π™Ώπ™΄πšπ™Έπ™Ύπ™³_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity32
Figure 5.50.7. Automaton for the 𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴 parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
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Figure 5.50.8. Automaton for the 𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity42
Figure 5.50.9. Hypergraph of the reformulation corresponding to the automaton of the 𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴 parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity51
Figure 5.50.10. Hypergraph of the reformulation corresponding to the automaton of the 𝙼𝙸𝙽_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity52
Figure 5.50.11. Automaton for the π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴 parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity61
Figure 5.50.12. Automaton for the π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity62
Figure 5.50.13. Hypergraph of the reformulation corresponding to the automaton of the π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_𝙲𝙷𝙰𝙽𝙢𝙴 parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity71
Figure 5.50.14. Hypergraph of the reformulation corresponding to the automaton of the π™Όπ™°πš‡_πš‚π™Έπš‰π™΄_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity72
Figure 5.50.15. Automata for the 𝙽𝙱_𝙲𝙷𝙰𝙽𝙢𝙴 and 𝙽𝙱_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ parameters of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity8
Figure 5.50.16. Hypergraph of the reformulation corresponding to the automaton of the 𝙽𝙱_𝙲𝙷𝙰𝙽𝙢𝙴 parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity91
Figure 5.50.17. Hypergraph of the reformulation corresponding to the automaton of the 𝙽𝙱_π™²π™Ύπ™½πšƒπ™Έπ™½πš„π™Έπšƒπšˆ parameter of the πšŒπš‘πšŠπš—πšπšŽ_πšŒπš˜πš—πšπš’πš—πšžπš’πšπš’ constraint
ctrs/change_continuity92